The Theory of Dynamic Hedging

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The Theory of Dynamic Hedging

• Nassim Nicholas Taleb
• Courant Institute of Mathematical Science
• Sept 4, 2003

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About this part of the course

• This part is Clinical Finance, which will be
  further defined in the next lecture.
• It is not the marriage of theory practice.
  Practice comes first last. This is best called
  theoretically inspired enhanced practice.
• No (or minimum) theorems, no proofs. The
  important matter is to be convinced. Why?
  Because theorems are only as good as the
  assumptions on which they are built.

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Some Holes With Existing Theory

• According to strict theoretical considerations,
  derivatives do not exist. Markets are
  fundamentally incomplete, and we have to live
  with it.
• Hakanssons paradox if markets are complete we
  do not need options if they are incomplete then
  according to financial theory we cannot price
  options
• The paradox has not been solved so far in finance
  theory ?? Finance theory may be total nonsense.
• The objective of this course is to make you live
  with it as well so you do not get a shock when
  you get out of here.
• Ask questions --the real world does not have an
  owners manual.

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• Clinical finance will be further discussed after
  a brief presentation of the existing theories
  --so we have enough material to engage in a
  critique of the current framework.

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The Theoretical Backbone of Modern Finance

• This first lecture will focus on the theoretical
  backbone of modern finance, particularly in what
  applies to asset pricing
• We will explore the origin of the thinking in
  financial economics
• If so little in successful quantitative Wall
  Street is linked to the financial economics
  aspect of finance, it is not quite without a
  reason

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Neoclassical Economics

• Adam Smiths invisible hand
• Walras auctioneer
• Marshalls partial equilibrium
• Arrow Debreus proof of the existence and
  uniqueness
• The central conclusion is the idea of
  laisser-faire the government should not
  interfere with the system of markets that
  allocates resources in the private sector of the
  economy

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Arrow-Debreu General Equilibrium

• A competitive system with market prices
  coordinates the otherwise independent activities
  of consumers and producers acting purely in their
  self-interest.
• Stands on shaky empirical foundations
• no information
• no adverse selection
• no intermediation, no transaction costs
• the model might have been reverse engineered,
  i.e. the correct assumptions were chosen because
  they led to the adequate solution.

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Some more attributes

• Theoretically Elegant
• Idealized

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Enters Uncertainty

• Arrow is credited with the introduction of
  uncertainty in the model, thanks to a contraption
  now called Arrow (now Arrow-Debreu)
  securities.
• In a 2-period model, it delivers 1 unit of
  numeraire in a given state of the world, 0
  otherwise.
• These securities complete the market, i.e.
  eliminate uncertainty as agents can buy them as
  insurance.

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Completeness

• Definition of a complete market

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State Prices

• a.k.a.state contingent claims, elementary
  securities, building blocks,
• These securities, by arbitrage, sum up to 1. Like
  probabilities, they are exhaustive and mutually
  exhaustive.
• It is important to see that they are not quite
  probabilities, even when translated into their
  continuous price time limit.
• This leads to the analog state price density for
  one period models.

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Lexicon

• State price a security that pays 1 in a state of
  the world, 0 elsewhere
• The price paid today for a state price resembles
  a density.
• Why resemble? Because of the difference between
  probability and pseudoprobability.
• Why pseudoprobability? Something called arbitrage

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Credits

• Note I credit for the exposition of the next
  three theorems, Rubinsteins e-textbook (1999),
  www.in-the-money.com
• Note a brief discussion of the inverse
  problem, i.e. the ability to pull out the state
  prices from derivatives (Breeden-Litzenberger,
  1978)

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First Theorem Existence

• Risk-neutral probabilities exist if and only if
  there are no riskless arbitrage opportunities.

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Arbitrage opportunities

• an arbitrage exists if and only if either
• (1) two portfolios can be created that have
  identical payoffs in every state but have
  different costs or
• (2) two portfolios can be created with equal
  costs, but where the first portfolio has at least
  the same payoff as the second in all states, but
  has a higher payoff in at least one state or
• (3) a portfolio can be created with zero cost,
  but which has a non-negative payoff in all states
  and a positive payoff in at least one state.

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Second Major Theorem Uniqueness

• The risk-neutral probabilities are unique if and
  only if the market is complete.
• Hint an incomplete market provides many
  solutions under this framework.

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Third Major Theorem Dynamic Completeness

• Arrow, in 1953, (tr. 1964) showed that, under
  some conditions, the ability to buy and sell
  securities can effectively make up for the
  missing securities and complete the market.

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Bachelier

• Aside from minor problems concerning the returns
  (arithmetic v/s logarithmic, which constitute a
  very small difference in common practice),
  Bachelier presented an option pricing tool that
  reposes on the actuarial distribution. In essence
  we are using his pricing method supplemented with
  arbitrage arguments.

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Keynes Arbitrage argument

• In 1923, Keynes effectively showed that by
  arbitrage argument, the forward needs to be equal
  to its arbitrage value, when lending borrowing
  are possible.
• Covered Interest Parity Theoremapplied to the
  forward for a currency pair and, by extension, to
  any security that can be lent and borrowed.

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Keynes Argument

• Currency 1 has a rate r1
• Currency2 has a rate r2
• Spot rate S
• Forward rate F
• F S (1r1)/(1r2)
• The Forward has nothing to do with
  expectations!!!

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The Importance of Keynes Intuition

• Keynes was the first person in modern times to
  express the notion that the forward is not an
  expected future price, but an arbitrage-derived
  pseudo-expectation.
• However it is of required use as an equivalent
  mean return in an arbitrage framework
• Arrows state prices are the equivalent
  probabilitites ?pseudoprobabilities

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The Essence of Black-Scholes-Merton

• What Black-Scholes did was not price options as
  we do it today. It merely made option pricing
  compatible with financial economics.
• There are two aspects to BSM
• First aspect the mean of the probability
  distribution used in their framework is that of
  the risk-neutral one (the µbecomes the difference
  between the carry and the financing).
• Second aspect There is no risk premium involved
  in the process --the package is deemed to be
  riskless.

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Assumptions Behind BSM

• no riskless arbitrage opportunities
• perfect markets
• constant r
• constant and known volatility (comment on the
  known)
• no jumps

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The Intuition

• Assuming an economy with no interest rates, the
  operator has two packages
• L1 short the European call option
• L2 long the local sensitivity of the option
  worth of stock
• L1 C(St,t) -C(St?t,t?t)
• L2 ? (St?t -St)

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• It is important to see that L1 and L2 are
  negatively correlated, that the negative
  correlation increases as ?t goes to 0
• It is important to see that, since the portfolio
  is delta neutral, there is no corresponding
  sensitivity of the total package to the asset
  price returns --therefore the return of the asset
  price becomes sensitive to the square variations
  .
• Pricing by replication allows the option to be a
  redundant security.

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Dynamic Hedging Effect
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Risk and Insurance

• Why dynamic hedging separates finance from other
  disciplines of risk bearing
• The option value is no longer the actuarial value
  of the payoff

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Food for Thought

• As an option trader I deem dynamic hedging
  unattainable --most of the package variance comes
  from the jumps. We cannot ignore the actuarial
  aspect of things.
• Black-Scholes reposes on a known distribution,
  with known parameters --I do not know much about
  the distribution

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The problem of the Normal/Lognormal

• Blaming Bachelier for having a normal not
  lognormal distribution may be unfair since in
  the real world we use a distribution of some
  uncharacterized shape

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Butterfly

• Buy call Struck at K ?
• Buy call Struck at K - ?
• Sell 2 calls struck at K
• What do we get at expiration?

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Breeden-Litzenberger

• The infinitely small butterfly scaled by 1/? at
  the limit delivers 1 at expiration at K and 0
  elsewhere
• More on that with JGs part the butterfly,
  called elementary securities, are the building
  block of everything